# Do all parameters have to have the same nature in a structural change test?

This is a duplicate of this query

(I was asked to shift the query from economics.stackexchange to Cross Validated SE.)

Lets say I am building a market model to estimate the beta of a stock with respect to a index of stocks.

The beta maybe Constant / Autoregressive of order 1 / doing a random walk. I realize there are time varying regression approaches for estimation of the above parameters IF I have an idea of the true nature of the parameters.

What is not clear to me if there are economic principles available to think about the nature of the parameters.

Should I assume that the intercept and slope have the same nature ( ie. they are both constant / Autoregressive of order 1 / doing a random walk.)

Is it possible that the intercept and slope have different characteristics ? Eg. the intercept is doing a random walk while the slope is AR1.

Can someone share the intuition behind this ? What should I be reading to understand this ?

• I don't think there's that much economic theory behind it. mostly, I've seen random walks because they're simple and also kind of implementing a bayesian prior in the sense that, my best bet of the future value, is today's value. Maybe read Hamilton's seminal paper on regime switching. There maybe something in there that helps you but I doubt it. – mlofton Apr 25 at 14:07
• Also, check out Rosenberg's paper in the 70's ( time varying regression coefficients ) because I think he came up with the concept. – mlofton Apr 25 at 14:08
• I can see what you are saying, the intuition behind choosing whether ONE variable is doing a RW / AR1. My main query is if we have MULTIPLE coefficients being modeleld say $y_t = a x_{1t} + b x_{2t} + \epsilon_t$ should they all have the same nature ? Or can they have different natures ? – user2338823 Apr 26 at 6:56
• Hi: I've never seen a random coefficient model where the coefficients didn't all have the same underlying model for how they move. In fact, I think assuming otherwise, would make estimation REALLY COMPLEX. I would stick to the same underlying distribution for all the coefficients. – mlofton Apr 27 at 14:19
• Hi: you may want to look at Andrew Harvey's blue book. (1990). I forget the ttitle but he has examples in there for estimating structural time series using the KF approach where one assumes various underlying models for the coefficients. Come to think of it, he may have an example or two in there, where the coeffs are AR(1) and maybe the other is RW. but, for the most part he uses examples where all the coeffs are RW's. It's a dense but good book and I recommend it, particularly for what you're doing. – mlofton Apr 27 at 14:24